Over at avforums.com I've heard a while ago that a user was told by the BBC tech support that the app has all those limitations because of the lack of "MPEG-DASH" support on Apple TV.
So the solution would be that the BBC does some extra work to find workarounds for Apple TV or Apple would have to add support for "MPEG-DASH" in tvOS. I don't think any of this will happen.
I think this is similiar to Netflix not supporting interactive content on Apple TV because they say there is no support for "seamless branching". Or Spotify not rolling out their new "TV experience" to Apple TV because they can't use the same cross-platform code.
So to me it seems that we can also thank Apple for the state of some apps. Not supporting some standards and technologies in tvOS that are supported on many other devices. And I think it's understandable that some app providers don't want to do time and cost intensive extra work for the Apple TV device that has a relatively low market share.